1915] NOTE ON THE FORMULA. FOR THE GRADIENT WIND 277
dv dv dv „ ,,
_ • +u- -- f- fl _ -|- 2 6)16 — CO-?/*.
at dx dy J
Since the relative motion is supposed to be steady, du/dt, dv/dt disappear, and the dynamical equations are
1 dp 2l0 du du n.
— f- = &ro3 + 2wv —u-^ -- v -T~ . ..................... (1)
p dx dx dy
I dp dv dv /0.
-- r- = ft)2?/ — 2d)U — U^j -- V -r- ...................... (2)
p dy J dx dy N
The velocities u, v may be expressed by means of the relative stream-function -\/r :
u = dtyjdy, v —
Equations (1), (2) then become
1 4 *
_i*J(**)V*m+vv.^, ...... W
2 cfcc (V<&»/ \dyJ } r dx
-+ + V^.; ...... (4)
p dy dy 2 dy \dxj \dy) f Y rfy
and on integration, if we leave out the part of p independent of the relative motion,
in which by a known theorem V2-v|r is a function of ^ only. If <a be omitted, (5) coincides with the equation given long ago by Stokes f. It expresses p in terms of •x/r ; but it does not directly allow of the expression of -\|r in terms of p, as is required if the data relate to a barometric chart.
We may revert to the more usual form, if in (1) or (3) we bake the axis of x perpendicular to the direction of (relative) motion at any point. Then u = 0, and
1 dp dty dty /ftN
- -/- = 2wv + -3- —^ ......................... (6)
p dx dx dy*
But since dty/dy = 0, the curvature at this place of the stream-line (^ = const.)
^ . ~ r " dyz ' dx'
and thus — f = 2cov + — , .............................. (7)
p dx ~ r
* Lamb's Hydrodynamics, § 206i
f Carnb. Phil. Trans. Vol. vn. 1812 ; Math, and, Phys. Papers, Vol. i. p. 9.t yet been proved theoretically that the stability can fail for finite disturbances on the supposition of perfectly smooth walls ; but such failure seems probable. We know from the work of Keynolds, Lorentz, and Orr that no failure of stability can occur unless @D*jv > 177, where D is the distance between, the walls, so that j3D represents their relative motion.